Enter a value, the mean, and the standard deviation. You'll get the z-score — how many standard deviations the value sits from the mean — plus its percentile in a normal distribution.
How the z-score calculator works
A z-score re-expresses a raw value on a universal scale: distance from the mean, measured in standard deviations. That's what makes it powerful — a 130 on an IQ test and a 700 on the SAT math section look nothing alike, but both are z ≈ +2, so they represent roughly the same rarity. The calculator subtracts the mean from your value, divides by the standard deviation, and then uses the standard normal distribution to translate that z-score into a percentile.
The formula
z = (x − μ) ÷ σ
Here x is your raw value, μ is the mean of the distribution, and σ is its standard deviation (which must be greater than zero — a distribution with no spread has no meaningful z-scores).
Worked example
IQ scores are designed with a mean of 100 and a standard deviation of 15. For a score of 130:
z = (130 − 100) ÷ 15 = 30 ÷ 15 = +2
A z-score of +2 lands at about the 97.7th percentile — only around 2.3% of people score higher. For comparison, a score of 115 gives z = +1, which is about the 84.1st percentile.
The 68–95–99.7 rule
For anything roughly bell-shaped, z-scores map to rarity in a way worth memorizing: about 68% of values fall within z = ±1, about 95% within ±2, and about 99.7% within ±3. So a z-score beyond ±2 puts a value in the most unusual 5% of the distribution, and beyond ±3 in the most unusual 0.3%. One caveat: this mapping assumes normality. Income, house prices, and social media follower counts are all heavily skewed, and for data like that a z-score of +3 can be far less rare than the rule suggests.
Frequently asked questions
What is a z-score?
A z-score expresses how far a value sits from the mean, measured in standard deviations. A z-score of 0 means the value equals the mean, +2 means two standard deviations above it, and −1.5 means one and a half below. It's a universal ruler that lets you compare values from completely different scales.
How do I convert a z-score to a percentile?
Look up the z-score in the standard normal distribution (this calculator does it for you). A z-score of 0 is the 50th percentile, +1 is about the 84th, +2 about the 98th, and −1 about the 16th. The percentile tells you what share of a normally distributed population falls below your value.
What is a good z-score?
It depends entirely on context — a high z-score is great for a test result and terrible for blood pressure. As a rule of thumb, values within ±1 are ordinary (about 68% of a normal population), beyond ±2 are notably unusual (outer 5%), and beyond ±3 are rare (outer 0.3%).
Can a z-score be negative?
Yes — a negative z-score simply means the value is below the mean. A z-score of −2 is exactly as far from the mean as +2, just on the low side. The sign tells you the direction; the size tells you the distance.
Does a z-score require a normal distribution?
Computing the z-score itself doesn't — it's just (value − mean) ÷ standard deviation, valid for any data. But converting it to a percentile does assume the data is roughly normal. If your distribution is heavily skewed, the percentile this calculator reports won't match your data's actual percentile.